The dogma that mathematical proofs could, in principle, be turned into logical proofs remains a conjecture; it is not feasible in practice. Existing proof systems are not up to the job; can they be beefed up sufficiently? Research community disconnect!https://xenaproject.wordpress.com/2020/02/09/where-is-the-fashionable-mathematics/ …
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Certainly I'd say the goal of a proof is to write things you're confident you _could_ turn into formal logical statements if you needed to. We're too lazy to do the dereferencing, but we're pretty sure we could.
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Of course, we could easily be wrong about that, and almost assuredly are wrong at least some of the time. But our intention is to write only things that we could turn into logical proofs.
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On second thought maybe this is true if one would accept a reformulation of the above statement as "Math is (mostly) logically sound"
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